#8933162

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#49206	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4133 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2)(3,9)(4,8)(5,6)(7,11)(10,16)(12,15)(13,14)(17,18,22,25)(19,26,29,23)(20,24,31,21)(27,32,30,28)(33,34,38,40)(35,39,43,48)(36,44,46,41)(37,47,45,42);(1,3)(2,7)(4,12)(5,9)(6,11)(8,16)(10,13)(14,15)(17,19,22,29)(18,23,25,26)(20,30,31,27)(21,28,24,32)(33,35,38,43)(34,39,40,48)(36,45,46,37)(41,47,44,42);(2,5)(4,13)(7,9)(10,12)(20,31)(21,24)(27,30)(28,32)(34,40)(35,43)(36,46)(42,47);(1,4,6,13)(2,8,5,14)(3,10,11,12)(7,15,9,16)(17,20)(18,24)(19,27)(21,25)(22,31)(23,28)(26,32)(29,30)(33,36,38,46)(34,41,40,44)(35,37,43,45)(39,42,48,47);(1,3)(2,7)(4,12)(5,9)(6,11)(8,16)(10,13)(14,15)(17,19)(18,23)(20,30)(21,28)(22,29)(24,32)(25,26)(27,31)(33,37)(34,42)(35,36)(38,45)(39,41)(40,47)(43,46)(44,48);(1,5,6,2)(3,9,11,7)(4,14,13,8)(10,16,12,15)(17,21,22,24)(18,20,25,31)(19,28,29,32)(23,30,26,27)(33,34)(35,39)(36,41)(37,42)(38,40)(43,48)(44,46)(45,47)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8840358);